upgrade your kernel, your card's HDMI isn't supported in that version.
go atleast 3.2.0_________________Only two things are infinite, the universe and human stupidity and I'm not sure about the former - Albert Einstein
ProjectFootball

your card is an r8xx chip, this is supported under the r7xx chip, as you can see from your own print, r7xx is not in the list.

beside that, your kernel is obsolete and I find no reason why someone will stay with that kernel version unless it is production server which is not your case from the question you are asking._________________Only two things are infinite, the universe and human stupidity and I'm not sure about the former - Albert Einstein
ProjectFootball

I have a motherboard with embedded graphics - I believe HD4200, and I believe that is covered by the r600 driver. But I don't believe I've ever seen names as fine-grained as you show, nor do I know the query which would return such. To be fair, I should really look at /var/log/Xorg.0.log more carefully, but I don't believe I've seen names like that, there. (With the CHIP_ prefix, in particular.)_________________.sigs waste space and bandwidth

Madison - I use, there one GPU.
Wrestler - decided purchase. I guess there one gpu too.

how many gpus you have currently on your system?

"Wrestler" is the codename for embedded graphics on AMD E-350/450 APUs. So, two GPUs (a 6310 and a discrete 5730 - seems odd that such a system would have a "downgrade" to the previous gen installed separately).